I always found $\mathrm{Tor}_R\left(M,N\right) \cong \mathrm{Tor}_R\left(N,M\right)$ for a commutative ring $R$ and two $R$-modules $M$ and $N$ to be mysterious. Then again I have no idea about homology and thus wouldn't be surprised if this is a triviality from an appropriate viewpoint.

Volker Strehl's generalized cyclotomic identity (Corollary 6 in Volker Strehl, *Cycle counting for isomorphism types of endofunctions* states that $\prod\limits_{k\geq 1} \left(\dfrac{1}{1-az^k}\right)^{M_k\left(b\right)} = \prod\limits_{k\geq 1}\left(\dfrac{1}{1-bz^k}\right)^{M_k\left(a\right)}$ in the formal power series ring $\mathbb Q\left[\left[z,a,b\right]\right]$, where $M_k\left(t\right)$ denotes the $k$-th necklace polynomial $\dfrac{1}{k}\sum\limits_{d\mid k} \mu\left(d\right) t^{k/d}$. I recall this being not particularly difficult, but quite useful.

Every nontrivial commutativity of some family of operators probably qualifies as an unexpected symmetry. Here are three examples:

**1.** Consider the group ring $\mathbb Z\left[S_n\right]$ of the symmetric group $S_n$. For every $i\in \left\{1,2,...,n\right\}$, define an element $Y_i \in \mathbb Z\left[S_n\right]$ by $Y_i = \left(1,i\right) + \left(2,i\right) + ... + \left(i-1,i\right)$ (a sum of $i-1$ transpositions). Then, $Y_i Y_j = Y_j Y_i$ for all $i$ and $j$ in $ \left\{1,2,...,n\right\}$. This is a simple exercise, and the $Y_i$ are called the *Young-Jucys-Murphy elements*.

**2.** Consider the group ring $\mathbb Z\left[S_n\right]$ of the symmetric group $S_n$. For every $i\in \left\{0,1,...,n\right\}$, define an element $\mathrm{Sch}_i \in \mathbb Z\left[S_n\right]$ as the sum of all permutations $\sigma \in S_n$ satisfying $\sigma\left(1\right) < \sigma\left(2\right) < ... < \sigma\left(i\right)$. (Note that $\mathrm{Sch}_0 = \mathrm{Sch}_1$ when $n\geq 1$.) Then, $\mathrm{Sch}_i \mathrm{Sch}_j = \mathrm{Sch}_j \mathrm{Sch}_i$ for all $i$ and $j$ in $ \left\{0,1,...,n\right\}$. In fact, $\mathrm{Sch}_i \mathrm{Sch}_j = \sum\limits_{k=0}^{\min\left\{n,i+j-n\right\}} \dbinom{n-j}{i-k} \dbinom{n-i}{j-k} \left(n+k-i-j\right)! \mathrm{Sch}_k$, which makes the symmetry maybe not that surprising (no similar equalities hold in cases **1** and **3**!). See Manfred Schocker, *Idempotents for derangement numbers*, Discrete Mathematics, vol. 269 (2003), pp. 239-248 for a proof. (This is also proven in my answers to Is this sum of cycles invertible in QSn? now, except that instead of the condition $\sigma\left(1\right) < \sigma\left(2\right) < ... < \sigma\left(i\right)$ I require $\sigma\left(n-i+1\right) < \sigma\left(n-i+2\right) < ... < \sigma\left(n\right)$ in that thread. But the two conditions can be transformed into one another by the automorphism $S_n \to S_n,\ \sigma \mapsto w \circ \sigma \circ w$ of $S_n$, where $w$ is the permutation in $S_n$ that sends each $i$ to $n+1-i$.)

**3.** Consider the group ring $\mathbb Z\left[S_n\right]$ of the symmetric group $S_n$. For every $i\in \left\{1,2,...,n\right\}$, define an element $\mathrm{RSW}_i \in \mathbb Z\left[S_n\right]$ as

$\sum\limits_{1\leq u_1 < u_2 < ... < u_i\leq n} \sum\limits_{\substack{\sigma\in S_n, \\ \sigma\left(u_1\right) < \sigma\left(u_2\right) < ... < \sigma\left(u_i\right)}} \sigma$.

Then, $\mathrm{RSW}_i \mathrm{RSW}_j = \mathrm{RSW}_j \mathrm{RSW}_i$ for all $i$ and $j$ in $ \left\{1,2,...,n\right\}$. This is Theorem 1.1 in Victor Reiner, Franco Saliola, Volkmar Welker, *Spectra of Symmetrized Shuffling Operators*, arXiv:1102.2460v2, and a nice proof remains to be found.

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